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Thank you Peter, however I'm not interested in this identity per se (I described it in a blog post more than ten years ago as one of the motivations for choosing B(1) = 1/2) or in its proof, which Ira Gessel gave here on MO. My concern is exactly what the request in the last line says.
Following on Ira Gessel's comments one might note that this is the simplest case of the super ballot numbers. A generalization of the integral formula is given in OEIS A135573.
